Harmonic Number

Author

Eric W. Weisstein
May 13, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/HarmonicNumber.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HarmonicNumber.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Usage

?HarmonicNumber

Enumeration

HarmonicNumber[Range[10]]

Powers of 10

Conjectured

N[Log[10,E],20]

Series

expr=Simplify[Normal[Series[Log[10,HarmonicNumber[x]],{x,∞,1}]]/.x->10^n,n∈Integers&&n>0]

Table[1+Floor[expr],{n,0,20}]

Enumeration

v5.2

Transpose[Table[Digits/@Through[{Numerator,Denominator}[HarmonicNumber[10^n]]],{n,0,7}]]//Timing

Transpose[Table[Digits/@Through[{Numerator,Denominator}[HarmonicNumber[10^n]]],{n,0,8}]]//Timing

v6

Transpose[Table[Digits/@Through[{Numerator,Denominator}[HarmonicNumber[10^n]]],{n,0,7}]]//Timing

Transpose[Table[Digits/@Through[{Numerator,Denominator}[HarmonicNumber[10^n]]],{n,0,8}]]//Timing

...crashes on Mac...

Transpose[Table[Digits/@Through[{Numerator,Denominator}[HarmonicNumber[10^n]]],{n,0,9}]]//Timing

Primes

Numerators

Position[Numerator/@HarmonicNumber[Range[1000]],_?PrimeQ]//Flatten//Timing

Position[Numerator/@HarmonicNumber[Range[5000]],_?PrimeQ]//Flatten//Timing

l={2,3,5,8,9,21,26,41,56,62,69,79,89,91,122,127,143,167,201,230,247,252,290,349,376,459,489,492,516,662,687,714,771,932,944,1061,1281,1352,1489,1730,1969,2012,2116,2457,2663,2955,3083,3130,3204,3359,3494,3572,3995,4155,4231,4250,4496,4616,5069,5988,6656,6883,8067,8156,8661,9097,9375,9977,10702,12315,12519,12728,13152,13571,14009,14247,14479,15299,17366,19645,21881,22307,25248,27754,32224,32725,34620,36282,39538,51708,60620,62849,63942,69294,69927,77449,78128,78993,81658};

ListPlot[l,PlotRange->All]

ListPlot[Log[l]^2,PlotRange->All,PlotStyle->Red,AxesLabel->{n,Log[HarmonicNumber[n]]^2}]

Length[l]

t={#,IntegerLength[Numerator[HarmonicNumber[#]]]}&/@l

MapIndexed[{#2[[1]],#1}&,l]//TableForm

Do[
Print[{n,l[[n]],PrimeQ[Numerator[HarmonicNumber[l[[n]]]]]//Timing}],
{n,Length[l]}]

Denominator

Position[Denominator/@HarmonicNumber[Range[10^4]],_?PrimeQ]//Flatten//Timing

Position[Denominator/@HarmonicNumber[Range[10^5]],_?PrimeQ]//Flatten//Timing

Denominator[HarmonicNumber[Range[20]]]

Plot

<<Utilities`Plot`
<<Utilities`Typesetting`

IntegerListPlot[HarmonicNumber/@Range[50],AxesLabel->TraditionalForm/@{n,HarmonicNumber[n]}]

ComplexPlot3D[HarmonicNumber[z],{z,-5,5},PlotPoints->150]//Timing

ComplexContourPlot[HarmonicNumber[z],{z,-5,5}]//Timing

Recurrence Equation

RSolve[{H[n]==1/n+H[n-1],H[1]==1},H[n],n]

FullSimplify[%]

In Terms of Polygamma

FunctionExpand[HarmonicNumber[n]]

Alternating Harmonic Number

(∑_k^n)((((-1))^(k+1))/(k^r))==(2^(-r)) ((((-1))^n) (ζ(r,(n/2)+1)-ζ(r,((n+1)/2)))+((2^r)-2) ζ(r))

AlternatingHarmonicNumber[n_,r_]:=Sum[(-1)^(k+1)/k^r,{k,n}]

Sum[(-1)^(k+1)/k^r,{k,n}]

AlternatingHarmonicNumber[n,r]//FullSimplify

(∑_k^n)((((-1))^(k+1))/k)==(1/2) ((((-1))^n) ((Subscript[H, (n-1)/2])-(Subscript[H, n/2]))+2 ln(2))

V6

Sum[((((-1))^(k+1))/k),{k,n}]

FullSimplify[%]

V7

Sum[((((-1))^(k+1))/k),{k,n}]

FunctionExpand[%]

FullSimplify[%]

FullSimplify[Subtract@@{(1/2) ((((-1))^n) (HarmonicNumber[(1/2) (-1+n)]-HarmonicNumber[(n/2)])+Log[4]),-(((-1))^n) LerchPhi[-1,1,1+n]+Log[2]}]

FunctionExpand[Subtract@@{(1/2) ((((-1))^n) (HarmonicNumber[(1/2) (-1+n)]-HarmonicNumber[(n/2)])+Log[4]),-(((-1))^n) LerchPhi[-1,1,1+n]+Log[2]}]

Table[(1/2) (E^(I n π)) (HarmonicNumber[(1/2) (-1+n)]-HarmonicNumber[(n/2)]+2 LerchPhi[-1,1,1+n])/.n->RandomComplex[],{100}]//Chop//Union

(∑_k^(2 n))((((-1))^(k+1))/k)

Sum[((((-1))^(k+1))/k),{k,2n}]

FunctionExpand[%]

Simplify[-(((-1))^(2 n)) LerchPhi[-1,1,1+2 n]+Log[2]-(HarmonicNumber[2n]-HarmonicNumber[n])]

Power Harmonic Numbers

(s=Table[{FunctionExpand[HarmonicNumber[n,k]],(1/n^k+1/(k-1)!PolyGamma[k-1,n]+Zeta[k])},{k,1,9,2}])//ColumnForm

Subtract@@@s

Sums

(∑_(n=1)^∞)((Subscript[H, n])/(n (2^n)))==((π^2)/12)

Math. Mag. 60, pp. 118–119, 1987.

Sum[HarmonicNumber[n]/(n 2^n),{n,∞}]

(∑_(n=0)^∞)(((z^n) (Subscript[H, n]))/(n!))==(e^z) (Γ(0,z)+log(z)+ℽ)

Sum[(((z^i)HarmonicNumber[i])/(i!)),{i,0,∞}]

(∑_(n=1)^∞)((Subscript[H, k])/k)==(1/12) (6 (Subscript[H, n]^2)-6 (ψ^((1)))(n+1)+(π^2))

Series[FunctionExpand[HarmonicNumber[n]],{n,∞,3}]

v5.2

Sum[((HarmonicNumber[k])/k),{k,n}]

Sum[((HarmonicNumber[k])/k),{k,∞}]

v6

Sum[((HarmonicNumber[k])/k),{k,n}]

Sum[((HarmonicNumber[k])/k),{k,∞}]

V7

Sum[((HarmonicNumber[k])/k),{k,n}]

FullSimplify[%]

Sum[((HarmonicNumber[k])/k),{k,∞}]

(∑_(n=1)^∞)((Subscript[H, n]^2)/(n^2))==((17 ζ(4))/4)

Sum[((HarmonicNumber[n]^2)/(n^2)),{n,∞}]

(∑_(n=1)^∞)((Subscript[H, n]^2)/(((n+1))^2))==((11 ζ(4))/4)

Sum[((HarmonicNumber[n]^2)/((n+1)^2)),{n,∞}]

(∑_(n=1)^∞)((Subscript[H, n])/(n^2))==2 ζ(3)

Sum[((HarmonicNumber[n])/(n^2)),{n,∞}]

(∑_(n=1)^∞)((Subscript[H, n])/(n^3))==((π^4)/72)

Sum[((HarmonicNumber[n])/(n^3)),{n,∞}]

9 (Subscript[H, 8]^((n)))-19 (Subscript[H, 9]^((n)))+10 (Subscript[H, 10]^((n)))+(∑_(k=1)^(n-1))(Subscript[H, 8]^((n-k))) (Subscript[H, 9]^((k)))-(∑_(k=1)^(n-1))(Subscript[H, 9]^((n-k))) (Subscript[H, 9]^((k)))-(∑_(k=1)^(n-1))(Subscript[H, 8]^((n-k))) (Subscript[H, 10]^((k)))+(∑_(k=1)^(n-1))(Subscript[H, 9]^((n-k))) (Subscript[H, 10]^((k)))==0

Michael Trott

FullSimplify[9*HarmonicNumber[8,n]-19*HarmonicNumber[9,n]+10*HarmonicNumber[10,n]+Sum[HarmonicNumber[8,n-k]*HarmonicNumber[9,k],{k,n-1}]-Sum[HarmonicNumber[9,n-k]*HarmonicNumber[9,k],{k,n-1}]-Sum[HarmonicNumber[8,n-k]*HarmonicNumber[10,k],{k,n-1}]+Sum[HarmonicNumber[9,n-k]*HarmonicNumber[10,k],{k,n-1}]]

ζ(5)==(1/11) (-16) (∑_(n=1)^∞)(((2 (((-1))^n)+1) (Subscript[H, n]))/(n^4))

Huvent

Zeta[5]//N

v5

-(16/11)Sum[((2(((-1))^n)+1)/(n^4))HarmonicNumber[n],{n,∞}]//Timing

N[%]

-(16/11)NSum[((2(((-1))^n)+1)/(n^4))HarmonicNumber[n],{n,∞}]//Chop//Timing

v6

-(16/11)Sum[((2(((-1))^n)+1)/(n^4))HarmonicNumber[n],{n,∞}]

FullSimplify[%]

N[%]

-(16/11)NSum[((2(((-1))^n)+1)/(n^4))HarmonicNumber[n],{n,∞}]

v7

-(16/11)Sum[((2(((-1))^n)+1)/(n^4))HarmonicNumber[n],{n,∞}]

(∑_(n=1)^∞)(z^n) (Subscript[H, n]^((r)))==-(((Subscript[Li, r])(z))/(z-1))

V6

Sum[HarmonicNumber[n,r]z^n,{n,∞}]

ByteCount[%]

V7

Sum[HarmonicNumber[n,r]z^n,{n,∞}]

ByteCount[%]

FullSimplify[%%]

2 (∑_(i=1)^∞)((Subscript[H, i]^((r)))/(i^r))==((ζ(r))^2)+ζ(2 r)

V6

2Sum[HarmonicNumber[i,r]/i^r,{i,Infinity}]

Subtract@@@Table[{2NSum[HarmonicNumber[i,r]/i^r,{i,Infinity},WorkingPrecision->19],Zeta[r]^2+Zeta[2r]//N},{r,2,5}]//Chop

Table[Zeta[r]^2+Zeta[2r],{r,2,4}]

V7

2Sum[HarmonicNumber[i,r]/i^r,{i,Infinity}]

(∑_(k=1)^∞)((Subscript[H, k]^((1)))/(k (((z+1))^k)))==-(Subscript[Li, 2])(-(1/z))

V6

Sum[((HarmonicNumber[k,1])/(k(((z+1))^k))),{k,∞}]//FullSimplify

FullSimplify[Subtract@@{
-((π^2)/6)-Log[-(1/z)] Log[(z/(1+z))]+PolyLog[2,1+(1/z)],
-PolyLog[2,-1/z]
}]

Subtract@@@Table[Log[1+(1/z)]+Log[(z/(1+z))]/.z->Random[Complex,{-(I+1),(I+1)}5],{20}]//Chop//Union

V7

Sum[((HarmonicNumber[k,1])/(k(((z+1))^k))),{k,∞}]

(∑_(k=1)^∞)((Subscript[H, k]^((1)))/(k (ϕ^(2 k))))==((π^2)/15)-(1/2) (((csch^(-1))(2))^2)

V6

Sum[((HarmonicNumber[k,1])/(k (GoldenRatio^(2k)))),{k,∞}]

FullSimplify[%]

FullSimplify[((π^2)/15)-2 ArcCoth[(Sqrt[5])] ArcCsch[2]+((3 ((ArcCsch[2])^2))/2)]

FullSimplify[Subtract@@{((π^2)/15)-2 ArcCoth[(Sqrt[5])] ArcCsch[2]+((3 ((ArcCsch[2])^2))/2),Pi^2/15 - (1/2)*ArcCsch[2]^2}]

V7

Sum[((HarmonicNumber[k,1])/(k (GoldenRatio^(2k)))),{k,∞}]

FullSimplify[%]

FullSimplify[Subtract@@{(1/30) (2 (π^2)+15 ArcCsch[2] (3 ArcCsch[2]+Log[4]-2 Log[3+(Sqrt[5])])),Pi^2/15-(1/2)*ArcCsch[2]^2}]

(∑_(k=1)^∞)((Subscript[H, k]^((1)))/((k^2) (2^k)))==ζ(3)-(1/12) (π^2) ln(2)

V6

Sum[((HarmonicNumber[k,1])/((k^2)(2^k))),{k,∞}]//FullSimplify

V7

Sum[((HarmonicNumber[k,1])/((k^2)(2^k))),{k,∞}]

(∑_(k=1)^∞)((Subscript[H, k]^((2)))/(k (2^k)))==((5 ζ(3))/8)

V6

Sum[((HarmonicNumber[k,2])/(k (2^k))),{k,∞}]//FullSimplify

V7

Sum[((HarmonicNumber[k,2])/(k (2^k))),{k,∞}]//FullSimplify

(∑_(i=1)^∞)((Subscript[H, i]^((2)))/(i^4))==((ζ(3))^2)-((π^6)/2835)

V6

Sum[HarmonicNumber[i,2]/i^4,{i,1,Infinity}]

NSum[HarmonicNumber[i,2]/i^4,{i,∞},WorkingPrecision->30]+Pi^6/2835

N[Zeta[3]^2,20]

V7

Sum[HarmonicNumber[i,2]/i^4,{i,1,Infinity}]

(∑_(i=1)^∞)((Subscript[H, i]^((4)))/(i^2))==((37 (π^6))/11340)-((ζ(3))^2)

V6

Sum[HarmonicNumber[i,4]/i^2,{i,∞}]//Timing

37/11340

37/11340Pi^6 -Zeta[3]^2//N

Subtract@@{37/11340Pi^6 -Zeta[3]^2,NSum[HarmonicNumber[i,4]/i^2,{i,∞},WorkingPrecision->30]}

V7

Sum[HarmonicNumber[i,4]/i^2,{i,∞}]

(Subscript[H, 2])[Conway-Guy]==(∑_i^n)(Subscript[H, i])==(n+1) (Subscript[H, n])-n

Sum[HarmonicNumber[i],{i,n}]//FullSimplify

(Subscript[H, 3])[Conway-Guy]==(∑_k^n)(k+1) ((Subscript[H, k+1])-1)==(Binomial[n+2, n]) ((Subscript[H, n+2])-(Subscript[H, 2]))==(1/4) (n+1) (n+2) (2 (Subscript[H, n+2])-3)

Sum[(1+k) (-1+HarmonicNumber[1+k]),{k,n}]//FullSimplify

FullSimplify[Binomial[n+2,n](HarmonicNumber[n+2]-HarmonicNumber[2])]

(∑_n^∞)(1/(HarmonicNumber2[n, -1]))==2

HarmonicNumber[n,-1]//Expand

Sum[1/HarmonicNumber[n,-1],{n,∞}]//FullSimplify

(∑_(n=1)^∞)(1/(HarmonicNumber2[n, -2]))==18-24 log(2)

Alexander Povolotsky
July 26, 2022

HarmonicNumber[n,-2]//Expand

Sum[1/HarmonicNumber[n,-2],{n,∞}]//FullSimplify

FullSimplify[Subtract@@{18-12 Log[4],18-24 Log[2]}]

Double Sums

(∑_(i=0)^(k-1))(∑_(j=k)^n)(((((-1))^(i+j-1)) (({{n}, {i}})) (({{n}, {j}})))/(j-i))==(∑_(i=0)^(l-1))(((({{n}, {i}})))^2) ((Subscript[H, n-i])-(Subscript[H, i]))

Sum[((((-1))^(i+j-1))/(j-i))Binomial[n,i]Binomial[n,j],{i,0,k-1},{j,k,n}]

Sum[((Binomial[n,i])^2)(HarmonicNumber[n-i]-HarmonicNumber[i]),{i,0,k-1}]

Table[
{Sum[((((-1))^(i+j-1))/(j-i))Binomial[n,i]Binomial[n,j],{i,0,k-1},{j,k,n}],
Sum[((Binomial[n,i])^2)(HarmonicNumber[n-i]-HarmonicNumber[i]),{i,0,k-1}]},
{k,10},{n,k,10}]

(∑_(k=1)^∞)(∑_(j=1)^∞)(((Subscript[H, j]) ((Subscript[H, k+1])-1))/(j k (k+1) (j+k)))==(2/3) (π^2) ζ(3)-2 ζ(3)+2 ζ(5)-((2 (π^2))/3)

Bailey et al. 2006, pp. 273–274

Exact

<<Utilities`Simplify`

TranscendentalRecognize[0.9992228377643094725840752661960140925`10.007839840269344,{Zeta[2],Zeta[3],Zeta[5],Zeta[2]Zeta[3]}]//FullSimplify

Expand[%/.(π^2)->6HoldForm[Zeta[2]]]

Numeric

N[(2/3) ((π^2) (-1+Zeta[3])-3 Zeta[3]+3 Zeta[5])]

NSum[((HarmonicNumber[j](HarmonicNumber[k+1]-1))/(k j(k+1)(j+k))),{j,∞},{k,∞}]

NSum[((HarmonicNumber[j](HarmonicNumber[k+1]-1))/(k j(k+1)(j+k))),{j,25},{k,25}]

v5.2

Over j

Sum[((HarmonicNumber[j](HarmonicNumber[k+1]-1))/(k j(k+1)(j+k))),{j,∞}]//FullSimplify

Plot[(1/(6 (k^2) (1+k)))((-1+HarmonicNumber[1+k]) ((π^2)+3 HarmonicNumber[-1+k,2]+3 (((EulerGamma+PolyGamma[0,k]))^2))),{k,1,20},PlotStyle->Red,PlotRange->All]

NSum[(1/(6 (k^2) (1+k)))((-1+HarmonicNumber[1+k]) ((π^2)+3 HarmonicNumber[-1+k,2]+3 (((EulerGamma+PolyGamma[0,k]))^2))),{k,∞}]

NSum[(1/(6 (k^2) (1+k)))((-1+HarmonicNumber[1+k]) ((π^2)+3 HarmonicNumber[-1+k,2]+3 (((EulerGamma+PolyGamma[0,k]))^2))),{k,∞},WorkingPrecision->50]

N[%,20]

Sum[(1/(6 (k^2) (1+k)))((-1+HarmonicNumber[1+k]) ((π^2)+3 HarmonicNumber[-1+k,2]+3 (((EulerGamma+PolyGamma[0,k]))^2))),{k,∞}]

Over k

Sum[((HarmonicNumber[j](HarmonicNumber[k+1]-1))/(k j(k+1)(j+k))),{k,∞}]//FullSimplify

Table[(1/(6 (((-1+j))^2) (j^2)))(HarmonicNumber[j] (12+6 (j^2)+(π^2)-j (12+(π^2))+6 (-1+j) HarmonicNumber[-2+j]-3 (-1+j) ((HarmonicNumber[-2+j])^2)-3 (-1+j) HarmonicNumber[-2+j,2])),{j,2,10}]//Simplify

NSum[(1/(6 (((-1+j))^2) (j^2)))(HarmonicNumber[j] (12+6 (j^2)+(π^2)-j (12+(π^2))+6 (-1+j) HarmonicNumber[-2+j]-3 (-1+j) ((HarmonicNumber[-2+j])^2)-3 (-1+j) HarmonicNumber[-2+j,2])),{j,2,∞}]//Timing

(1/(6 (((-1+j))^2) (j^2)))(HarmonicNumber[j] (12+6 (j^2)+(π^2)-j (12+(π^2))+6 (-1+j) HarmonicNumber[-2+j]-3 (-1+j) ((HarmonicNumber[-2+j])^2)-3 (-1+j) HarmonicNumber[-2+j,2]))/.j->1

Plot[(1/(6 (((-1+j))^2) (j^2)))(HarmonicNumber[j] (6 (2+(-2+j) j)-(-1+j) (π^2)+6 (-1+j) HarmonicNumber[-2+j]-3 (-1+j) ((HarmonicNumber[-2+j])^2)-3 (-1+j) HarmonicNumber[-2+j,2])),{j,1,20},PlotStyle->Red]

v6

Sum[((HarmonicNumber[j](HarmonicNumber[k+1]-1))/(k j(k+1)(j+k))),{j,∞},{k,∞}]

Sum[((HarmonicNumber[j](HarmonicNumber[k+1]-1))/(k j(k+1)(j+k))),{k,∞},{j,∞}]

Sum[((HarmonicNumber[j](HarmonicNumber[k+1]-1))/(k j(k+1)(j+k))),{k,∞}]//FullSimplify

NSum[(1/(6 (((-1+j))^2) (j^2)))(HarmonicNumber[j] (12+6 (j^2)+(π^2)-j (12+(π^2))+6 (-1+j) HarmonicNumber[-2+j]-3 (-1+j) ((HarmonicNumber[-2+j])^2)-3 (-1+j) HarmonicNumber[-2+j,2])),{j,∞}]//Timing

Sum[((HarmonicNumber[j](HarmonicNumber[k+1]-1))/(k j(k+1)(j+k))),{j,∞}]//FullSimplify

NSum[(1/(6 (k^2) (1+k)))(-1+HarmonicNumber[1+k]) ((π^2)+3 HarmonicNumber[-1+k,2]+3 (((EulerGamma+PolyGamma[0,k]))^2)),{k,∞}]//Timing

Sum[(1/(6 (k^2) (1+k)))(-1+HarmonicNumber[1+k]) ((π^2)+3 HarmonicNumber[-1+k,2]+3 (((EulerGamma+PolyGamma[0,k]))^2)),{k,∞}]

V7

Sum[((HarmonicNumber[j](HarmonicNumber[k+1]-1))/(k j(k+1)(j+k))),{j,∞},{k,∞}]

TimeConstrained[Sum[((HarmonicNumber[j](HarmonicNumber[k+1]-1))/(k j(k+1)(j+k))),{k,∞},{j,∞}],60]//Timing

(1/2) (ζ(r,n+1)-ζ(r,n+(1/2))+2 (Subscript[H, n]^((r)))-(2^r) (Subscript[H, 2 n]^((r)))+((2^r)-2) ζ(r))

id=Subtract@@{HarmonicNumber[n,r],2^(r-1)(HarmonicNumber[2n,r]-AlternatingHarmonicNumber[2n,r])}//FullSimplify

Table[id/.Thread[{n,r}->Table[Random[Integer,{1,5}],{2}]],{30}]//FullSimplify//Union

(∑_(k=1)^n)(1/(k^2))==(3/2) (∑_(k=1)^n)(1/((k^2) (({{2 k}, {k}}))))+(1/2)(∑_(i=1)^n)(∑_(j=1)^n)(((i-1)! (j-1)!)/((i+j)!))

V6

sum1=Sum[(1/(k^2)),{k,n}]

sum2=(1/2)Sum[(((i-1)!(j-1)!)/((i+j)!)),{i,n},{j,n}]+(3/2)Sum[(1/((k^2)Binomial[2k,k])),{k,n}]//FullSimplify

FullSimplify[sum2,n>0&&n∈Integers]

FullSimplify[sum1-sum2,n>0&&n∈Integers]

Table[{Sum[(1/(k^2)),{k,n}],},{n,5}]

V7

(1/2)Sum[(((i-1)!(j-1)!)/((i+j)!)),{i,n},{j,n}]+(3/2)Sum[(1/((k^2)Binomial[2k,k])),{k,n}]

FullSimplify[%]

Table[(1/6) (-(3/(((1+n))^2))+(π^2)-((9 ((Gamma[1+n])^2) HypergeometricPFQ[{1,1+n,1+n},{(3/2)+n,2+n},(1/4)])/(Gamma[3+2 n]))-((3 HypergeometricPFQ[{1,1+n,1+n},{2+n,3+n},1])/(((1+n))^2))+3 Gamma[2+n] (2 ((Gamma[1+n])^2) HypergeometricPFQRegularized[{1,1+n,1+n},{2+n,3+2 n},1]+Gamma[2+n] (-HypergeometricPFQRegularized[{2,2+n,2+n},{3+n,4+n},1]+n Gamma[n] HypergeometricPFQRegularized[{2,2+n,2+n},{3+n,4+2 n},1]))-3 PolyGamma[1,2+n]),{n,5}]//Simplify

(1/2)Sum[(((i-1)!(j-1)!)/((i+j)!)),{j,n},{i,n}]+(3/2)Sum[(1/((k^2)Binomial[2k,k])),{k,n}]

FullSimplify[%]

Table[(1/6) (-(3/(((1+n))^2))+(π^2)-((9 ((Gamma[1+n])^2) HypergeometricPFQ[{1,1+n,1+n},{(3/2)+n,2+n},(1/4)])/(Gamma[3+2 n]))-((3 HypergeometricPFQ[{1,1+n,1+n},{2+n,3+n},1])/(((1+n))^2))+3 Gamma[2+n] (2 ((Gamma[1+n])^2) HypergeometricPFQRegularized[{1,1+n,1+n},{2+n,3+2 n},1]+Gamma[2+n] (-HypergeometricPFQRegularized[{2,2+n,2+n},{3+n,4+n},1]+n Gamma[n] HypergeometricPFQRegularized[{2,2+n,2+n},{3+n,4+2 n},1]))-3 PolyGamma[1,2+n]),{n,5}]//Simplify

Exceeding Integers

Ceiling[n/.FindRoot[HarmonicNumber[n]==23,{n,5},WorkingPrecision->50,AccuracyGoal->Infinity,MaxIterations->20]]

l=Ceiling[Table[n/.FindRoot[HarmonicNumber[n]==k,{n,5},MaxIterations->30,WorkingPrecision->50,AccuracyGoal->Infinity],{k,27}]]

N[HarmonicNumber[#],20]&/@l

N[HarmonicNumber[#],20]&/@(l-1)

Number of Terms in the CF

Length/@Table[ContinuedFraction[HarmonicNumber[10^n]],{n,5}]//Timing

Length[ContinuedFraction[HarmonicNumber[10^6]]]//Timing

h6=HarmonicNumber[10^6];//Timing

Length[ContinuedFraction[h6]]//Timing

h3=HarmonicNumber[10^7];//Timing

Length[ContinuedFraction[h3]]//Timing

h8=HarmonicNumber[10^8];//Timing

Length[ContinuedFraction[h8]]//Timing

N[12Log[2]/Pi^2,20]